We will first explain the concepts of relatively hyperbolic group and the Bowditch boundary. We will then give some interesting examples of groups whose boundaries embed in the two-sphere. The most prominent family of this type is the class of geometrically finite Kleinian groups. However, we show that there are lots of relatively hyperbolic groups with planar boundaries that are not virtually Kleinian. We formulate a conjecture about which groups with planar boundary are virtually Kleinian, and prove this in a certain case. This is joint work in progress with Chris Hruska.
